DIAGRAMMATIC CHARACTERISATION OF ENRICHED ABSOLUTE COLIMITS
RICHARD GARNER
Abstract. We provide a diagrammatic criterion for the existence of an absolute col- imit in the context of enriched category theory.
An absolute colimit is one preserved by any functor; the class of absolute colimits was characterised for ordinary categories by Par´e [4] and for enriched ones by Street [5]. For categories enriched over a monoidal category V or bicategory W, the appropriate colimits are the weighted colimits of [6], and Street’s characterisation is in fact one of the class of absolute weights: those weights ϕ such that ϕ-weighted colimits are preserved by any functor. This is different to Par´e’s result, which gives a diagrammatic characterisation of when a particular cocone is absolutely colimiting. In this note, we give a result in the enriched context which is closer in spirit to Par´e’s than to Street’s. This result is very useful in practice, but seems not to be in the literature; we set it down for future use.
1. The result
1.1. Background. We work in the context of bicategory-enriched category theory;
see [6], for example. W will denote a bicategory whose homs are locally small, complete and cocomplete categories, and which is biclosed, meaning that for each 1-cell A : x → y in W, the composition functors A⊗(–) : W(z, x) → W(z, y) and (–)⊗A : W(y, z) → W(x, z) have right adjoints [A, –] and hA, –i respectively.
A W-category A comprises a set ob A of objects; for each a ∈ ob A an object
a ∈ ob W, the extent of a; for each pair of objects a, b, a hom-object C(b, a) ∈ W(a, b);
and identity and composition 2-cells ι : Ia → C(a, a) and µ : C(c, b) ⊗ C(b, a) → C(c, a) sat- isfying the expected axioms. A W-profunctor M : A −7→ B is given by objects M (b, a) ∈ W(a, b) and action maps µ : B(b0, b) ⊗ M (b, a) ⊗ A(a, a0) → M (b0, a0) satisfying unital- ity and associativity axioms. A profunctor map M → M0: A −7→ B comprises maps M (b, a) → M0(b, a) compatible with the actions by A and B. The identity profunctor A : A −7→ A has components A(b, a) with action given by composition in A. For profunc- tors M : A −7→ B and N : B −7→ C with B small, the tensor product N ⊗B M : A −7→ C
The support of Australian Research Council Discovery Projects DP110102360 and DP130101969 is gratefully acknowledged.
Received by the editors 2014-09-30 and, in revised form, 2014-10-29.
Transmitted by Ross Street. Published on 2014-10-31.
2010 Mathematics Subject Classification: 18A30, 18D20.
Key words and phrases: Absolute colimits, enriched categories.
Richard Garner, 2014. Permission to copy for private use granted.c
775
has components given by coequalisers P
b,b0N (c, b) ⊗ B(b, b0) ⊗ M (b0, a) ⇒P
bN (c, b) ⊗ M (b, a) (N ⊗B M )(c, a) and actions by C and A inherited from N and M . Small W-categories, profunctors and profunctor maps comprise a bicategory W-Mod. There is a full embedding W → W-Mod sending X to the W-category X with one object ? with (?) = X and X(?, ?) = IX.
If A and B are W-categories, then a W-functor F : A → B comprises an extent- preserving assignation on objects, together with 2-cells C(b, a) → D(F b, F a) subject to two functoriality axioms. If F : A → C and G : B → C are W-functors then there is an induced profunctor C(G, F ) : A −7→ B with components C(G, F )(b, a) = C(Gb, F a) and action derived from the action of F and G on homs and composition in C.
Given profunctors M : A −7→ B, N : B −7→ C and L : A −7→ C with B small, a profunctor map u : N ⊗BM → L is said to exhibit M as [N, L] if every map f : N ⊗B K → L is of the form u ◦ (N ⊗Bf ) for a unique ¯¯ f : K → M ; while it is said to exhibit N as hM, Li if every f : K ⊗BM → L is of the form u ◦ ( ¯f ⊗B M ) for a unique ¯f : K → N .
Given ϕ : A −7→ B in W-Mod and a functor F : B → C, a ϕ-weighted colimit of F is a functor Z : A → C and profunctor map a : ϕ → C(F, Z) such that for each C ∈ C, the map
ϕ ⊗AC(Z, C)−−−→ C(F, Z) ⊗a⊗A1 AC(Z, C)−−−→ C(F, C)µ (1) exhibits C(Z, C) as [ϕ, C(F, C)]. A functor G : C → D preserves this colimit just when the composite ϕ → C(F, Z) → D(GF, GZ) exhibits GZ as a ϕ-weighted colimit of GF ; the colimit is absolute when it is preserved by all functors out of C. [5] proves that ϕ-weighted colimits are absolute if and only if ϕ admits a right adjoint in W-Mod.
Dually, given ψ : B −7→ A in W-Mod and a functor F : B → C, a ψ-weighted limit of F is a functor Z : A → C and map b : ψ → C(Z, F ) such that for each C ∈ C, the map
C(C, Z) ⊗Aψ −−−→ C(C, Z) ⊗1⊗Ab AC(Z, F )−−−→ C(C, F )µ
exhibits C(C, Z) as hψ, C(C, Z)i. Absoluteness of limits is defined as before; now every limit weighted by ψ : B −7→ A is absolute if and only if ψ has a left adjoint in W-Mod.
1.2. Theorem. Let ϕ : A −7→ B admit the right adjoint ψ : B −7→ A in W-Mod, and let F : B → C and Z : A → C be W-functors. There is a bijective correspondence between data of the following forms:
(a) A map a : ϕ → C(F, Z) exhibiting Z as a ϕ-weighted colimit of F ; (b) A map b : ψ → C(Z, F ) exhibiting Z as a ψ-weighted limit of F ;
(c) Maps a : ϕ → C(F, Z) and b : ψ → C(Z, F ) such that the following two squares commute in W-Mod(A, A) and W-Mod(B, B):
A η //
Z
ψ ⊗Bϕ
b⊗Ba
C(Z, Z)oo µ C(Z, F ) ⊗B C(F, Z)
ϕ ⊗Aψ ε //
a⊗Ab
B
F
C(F, Z) ⊗AC(Z, F ) µ //C(F, F ) . (2)
Proof . Suppose first given (a); consider the composite profunctor map
ϕ ⊗AC(Z, F )−−−→ C(F, Z) ⊗a⊗A1 AC(Z, F )−−−→ C(F, F ) .µ (3) Evaluating in the second variable at any a ∈ A yields the map (1) exhibiting C(Z, F a) as [ϕ, C(F, F a)]; it follows easily that (3) exhibits C(Z, F ) as [ϕ, C(F, F )]. Applying this universality to the composite ε ◦ F : ϕ ⊗Aψ → B → C(F, F ) yields a unique map b : ψ → C(Z, F ) making the right square of (2) commute; we must show that the left one does too. Arguing as before shows that
ϕ ⊗AC(Z, Z)−−−→ C(F, Z) ⊗a⊗A1 AC(Z, Z)−−−→ C(F, Z)µ (4) exhibits C(Z, Z) as [ϕ, C(F, Z)]. It thus suffices to show that the left square of (2) com- mutes after applying the functor ϕ ⊗A(–) and postcomposing with (4); which follows by a short calculation using commutativity in the right square and the triangle identities.
So from the data in (a) we may obtain that in (c), and the assignation is injective, since b is uniquely determined by universality of a and commutativity on the right of (2).
For surjectivity, suppose given a and b as in (c); we must show that a exhibits Z as a ϕ- weighted colimit of F , in other words, that for each C ∈ C, the map (1) exhibits C(Z, C) as [ϕ, C(F, C)], or in other words, that for each map f : ϕ ⊗AK → C(F, C), there is a unique map ¯f : K → C(Z, C) such that f = µ◦(a⊗Af ) : ϕ⊗¯ AK → C(F, Z)⊗AC(Z, C) → C(F, C).
For existence, we let ¯f be the composite
K ∼= A ⊗AK −−−→ ψ ⊗η⊗A1 B ϕ ⊗AK −−−→ C(Z, F ) ⊗b⊗Bf BC(F, C)−→ C(Z, C) ;µ (5) now rewriting with the right-hand square of (2) and using the triangle identities and F ’s preservation of units shows that f = µ ◦ (a ⊗Af ). For uniqueness, let g : K → C(Z, C)¯ also satisfy f = µ ◦ (a ⊗Ag). Substituting into (5) shows that ¯f is the composite
K ∼= A ⊗AK −−−→ ψ ⊗η⊗A1 Bϕ ⊗AK −−−−−→ C(Z, F ) ⊗b⊗Ba⊗Ag BC(F, Z) ⊗AC(Z, C) −→ C(Z, C) ;µ which by rewriting with the left square of (2) and using Z’s preservation of identities is equal to g. This proves the equivalence (a) ⇔ (c); now (b) ⇔ (c) follows by duality.
1.3. Examples. We first consider examples wherein W is the one-object bicategory cor- responding to a monoidal category V.
• Let V = Set, and let ϕ be the weight for splittings of idempotents. The result recovers the bijection, for an idempotent e : A → A, between: maps p : A → B coequalising e and 1A; maps i : B → A equalising e and 1A; and pairs (i, p) with pi = 1A and ip = e.
• Let V = Set∗, and let ϕ be the weight for an initial object. The result recovers the bijection in a pointed category between: initial objects; terminal objects; and objects X with 1X = 0X.
• Let V = Ab, and let ϕ be the weight for binary coproducts. The result recovers the bijection, for objects A, B in a pre-additive category, between: coproduct diagrams i1: A → Z ← B : i2; product diagrams p1: A ← Z → B : p2; and tuples (i1, i2, p1, p2) such that pjik = δik and i1p1+ i2p2 = 1Z.
• Let V = W -Lat, and let ϕ be the weight for J-fold coproducts (for J a small set). The result recovers the bijection, for objects (Aj : j ∈ J ) in a sup-lattice enriched category, between: coproduct diagrams (ij: Aj → Z)j∈J; product diagrams (pj: Z → Aj)j∈J; and families (ij)j∈J and (pj)j∈J with pjik = δjk and W
jijpj = 1Z.
• Let V = k-Vect for k a field of characteristic zero, let G be a finite group, and let ϕ : k −7→ kG be the trivial right kG-module k. By Burnside’s Lemma, ϕ has a right adjoint kG −7→ k given by the trivial left kG-module k. Now the result recovers the bijection, for a G-representation A in a k-linear category, between: maps p : A → Z exhibiting Z as an object of coinvariants of A; maps i : Z → A exhibiting Z as an object of invariants of A; and pairs of maps (i, p) with pi = 1Z and ip = |G|1 Σg∈Gg.
We conclude with two examples where W is a genuine bicategory.
• Let (C, j) be a subcanonical site, and let W denote the full sub-bicategory of Span(Sh(C))op on objects of the form C(–, X). To any prestack p : E → C over C, we may (as in [1]) associate a W-category with objects those of E , extents (a) = p(a), and hom-object from a to b given by the span C(–, pa) ← E (a, b) → C(–, pb) in Sh(C); here E (a, b)(x) is the set of all triples (f, g, θ) with f : pa ← x → pb : g in C and θ : f∗(a) → g∗(b) in Ex (note that E (a, b) is a sheaf by the prestack condition).
For any cover (fi: Ui → U )i∈I in C, we have a W-category R[f ] with object set I, extents (i) = Uiand hom-objects R[f ](j, i) = C(–, Uj) ← C(–, Uj×UUi) → C(–, Ui).
There is a profunctor ϕ : U −7→ R[f ] with components given by the spans ϕ(i, ?) = C(–, Ui) ← C(–, Ui) → C(–, U ). Writing ψ : R[f ] −7→ U for the reverse profunctor, it is not hard to see that ϕ a ψ (in fact they are adjoint pseudoinverse).
The result now says the following. Given a prestack p : E → C, a cover (fi: Ui → U ) in C, and a family of spans pij: ai ← aij → aj: qij in E whose legs are cartesian over the projections Ui ← Ui ×U Uj → Uj, there is a bijection between: cocones (hi: ai → a) in E over the fi’s that are colimiting for the diagram comprised of the pij’s and qij’s; universal objects a ∈ EU equipped with vertical maps fi∗(a) → ai fitting into double pullback squares
fi∗(a)
oo · //fj∗(a)
ai aijp
ij
oo qij //aj ;
and objects a ∈ EU equipped with a family of maps (hi: ai → a) cartesian over the fi’s. This generalises [6, Proposition 5.2(b)]1.
1The proposition numbering here is taken from the TAC reprint.
• Let W denote the bicategory whose objects are sets, and whose hom-category W(X, Y ) is the category of finitary functors Set/Y → Set/X; note that W(X, Y ) ' [Fam(Y ) × X, Set], where Fam(Y ) has as objects, finite lists of elements of Y , and as maps (y0, . . . , ym) → (z0, . . . , zn), functions f : [m] → [n] such that yi = zf (i). To any cartesian multicategory M (i.e., a Gentzen multicategory in the sense of [3]) we may associate a W-category M whose objects of extent X are X-indexed families of objects of M , and whose hom-object between families (ax)x∈X and (by)y∈Y is the presheaf
M((by), (ax))(y0, . . . , ym; x) = M (by0, . . . , bym; ax)
in [Fam(Y ) × X, Set]; reindexing along maps in Y makes use of the cartesianness of the multicategory structure. Composition and units in M follow from those in M . Given a finite set X = {x0, . . . , xn}, let ϕ : 1 −7→ X be the W-profunctor whose unique component is the representable y(x0, . . . , xn; ?) ∈ [Fam(X)×1, Set]. This has a right adjoint ψ : X −7→ 1 whose unique component is the presheaf Σx∈Xy(?; x) ∈ [Fam(1) × X, Set]. The result now establishes a bijection, for any finite family (a0, . . . , an) of objects in a cartesian multicategory M , between data of the fol- lowing three forms: first, an object a and a multimap i ∈ M (a0, . . . , an; a), com- position with which induces bijections between M (b0, . . . , bk, a, c0, . . . , c`; d) and M (b0, . . . , bk, a0, . . . , an, c0, . . . , c`; d); second, an object a and unary maps pj ∈ M (a; aj), composition with which establishes bijections between M (b0, . . . , bk; a) and ΠjM (b0, . . . , bk; aj); third, an object a and maps i and pj as above such that pj◦ i = πj ∈ M (a0, . . . , an; aj) and i ◦ (p0, . . . , pn) = 1a ∈ M (a; a). This generalises [2, Proposition 3.5].
References
[1] Betti, R., Carboni, A., Street, R., and Walters, R. Variation through enrichment. Journal of Pure and Applied Algebra 29, 2 (1983), 109–127.
[2] Garner, R. Lawvere theories, finitary monads and Cauchy-completion. Journal of Pure and Applied Algebra 218, 11 (2014), 1973–1988.
[3] Lambek, J. Multicategories revisited. In Categories in computer science and logic (Boulder, 1987), vol. 92 of Contemporary Mathematics. American Mathematical So- ciety, 1989, pp. 217–239.
[4] Par´e, R. On absolute colimits. Journal of Algebra 19 (1971), 80–95.
[5] Street, R. Absolute colimits in enriched categories. Cahiers de Topologie et Geom´etrie Diff´erentielle Cat´egoriques 24, 4 (1983), 377–379.
[6] Street, R. Enriched categories and cohomology. Quaestiones Mathematicae 6 (1983), 265–283. Republished as: Reprints in Theory and Applications of Categories 14 (2005).
Department of Mathematics, Macquarie University, NSW 2109, Australia Email: [email protected]
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